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Thursday, February 14, 2008

M Theory Lesson 158

In lecture 25, Baez looks at the functor +1 on the groupoid of finite sets. Recall that addition is the disjoint union of sets, so this functor takes a finite set and adds a one point set to it. Now we can define groupoidified creation and annhilation operators. First take the span shown on the left. Applying the zero homology functor will turn the arrow +1 into the (groupoidified) creation operator. Similarly, starting with the functor +1 on the left yields the annihilation operator. The commutation relation between $a$ and $a^{\dagger}$ follows from considering the number of ways to take things out of a set and put them back in again. Think about it. Note that the composition of spans (the pink arrows) is given by a (weak) pullback, which conveniently exists for this category. Pullbacks allow products of arbitrary numbers of creation and annihilation operators.

Back down inside FinSet, recall that a pullback of two subspace arrows $U$ and $V$ is the intersection $U \cap V$. But in the groupoid FinSet0 the only arrows are bijections, and such limit diagrams do not exist. The finite set with $n$ elements has no arrow connecting it to the finite set with $n + 1$ elements. Thus the functor +1 acts simply as a categorified successor arrow for the ordinals.

Although not a satisfactory quantum gravitational definition for creation and annihilation, this approach goes some way towards giving a canonical diagrammatic representation for these operators. As Carl has pointed out, the Feynman diagrams for density matrix QFT should have an interpretation as products of creation and annihilation operators, a la Schwinger.